See “Zero Sum” is a misnomer, shifting and rescaling of utility functions breaks formulations that simply ask to take a sum of payoffs, but we can rescue the concept to mean that all outcomes/strategies of the game are Pareto efficient.
“Positive sum” seems to be about Kaldor-Hicks improvement, an outcome that in principle admits a subsequent redistribution of resources that would turn the outcome into a Pareto improvement (over some original situation of “not playing”), but there is no commitment or possibly even practical feasibility to actually perform the redistribution. This hypothetical redistribution step takes care of comparing utilities of different players. A whole game/interaction/project would then be “positive-sum” if each outcome/strategy is equivalent via a redistribution to some hypothetical “outcome” that is a Pareto improvement over the status quo of not engaging in the game/interaction/project. In actuality, without the hypothetical redistribution step, some players can end up worse off.
Hi Vladimir, thanks for your input, it has been fascinating going down the rabbit hole of nuance regarding the term “zero-sum”.
I agree that the term is more accurately denoting “constant-sum”, I think this is generally implied by most people using it. There was the interesting “zero-sum” example in the linked article that veered away from “constant-sum” with asymmetrical payoffs, 100,0 or 0,1, meaning that depending on the outcome of the game the total sum would be different. This, to me disqualifies it from being called a zero-sum game, given the common understanding that zero-sum denotes constant-sum. The example seemed to solve the problem by conflating zero-sum and constant-sum and then proceeded to stick to a strict definition of zero-sum, which was confusing. But perhaps I just need to sit with it longer.
To your point about Kaldor-Hicks, yes I guess many positive-sum situations could be described in these terms but I’m really referring to something more general—any situation where the total sum payoff increases regardless of Pareto improvements or promised reimbursement by other means to any party left worse off. For instance if a left-wing government were to increase taxes on the wealth, not offering them any reimbursement, but rather doing this based on the mandate that comes with being democratically elected, then this policy might be positive-sum due to the fact that dollar-for-dollar money makes a bigger difference to a poor person than a rich person, due to diminishing returns on happiness.
I really appreciate your comments, and intend to continue exploring the nuances you’ve raised. I think for a primer on non-zero-sum games, particularly with a site that is focused on practical solutions in the real world rather than pure theory, the more accessible (perhaps less nuanced) definitions I’ve used are probably appropriate.
disqualifies it from being called a zero-sum game, given the common understanding that zero-sum denotes constant-sum
The point is that the preference order over lotteries characterized by a utility function doesn’t change if you multiply the utility function by a positive value or add a constant to it. Utility function u(x) makes exactly the same choices as utility function 2u(x). If we start with a constant sum-of-utilities game (for two players) and then rescale one of the utilities, the sum will no longer be constant, but the game is still the same. You’d need to take a weighted sum instead to compensate for this change of notation. So the characterization of a game as “constant sum” doesn’t make sense if taken literally, since it doesn’t survive a mere change of notation that doesn’t alter anything about the actual content of the game.
I’m not sure how the game is the same when you add a constant. The game as proposed in the example is clearly different. I can see that multiplication makes no difference, and as such doesn’t make the sum non-constant. I don’t see how asymmetrically changing the parameters is a “mere change in notation”.
By the way, I’m sure you’re entirely correct about this, I just simply don’t see how there is a problem with using the concept of zero-sum understood as constant-sum.
Utility functions are a way of characterizing preference orderings between events. If a preference ordering satisfies certain properties, then there exists a utility function such that its expected value over the events can be used to decide which events are preferred over which other events (see VNMtheorem). Utility values are not defined with respect to anything else, they are not money or happiness or resources. In particular, utilities of different players can’t be compared a priori, without bringing in more structure (for example redistribution of resources in the setup of Kaldor-Hicks improvement establishes a way of comparing utilities of players, see the original comment).
If you add a constant to a utility function, its expected value over some event increases by the same constant. So if one event had greater expected utility than another, it would still be the case after you add the constant. This is the sense in which adding constants or multiplying by positive factors makes no difference.
Okay, so I think I get you now, in the imbalanced game, if the payoff is 100 or 1 as in “Zero Sum” is a misnomer, a rational player will still make the same decision, regardless of the imbalance with the other player, given the resulting preference ordering.
However while this imbalance makes no difference to the players’ decisions, it does make a difference to the total payoff, making it non-zero-sum. I’m having difficulty understanding why values such as happiness or resources cannot be substituted for utility—surely at some point this must happen if game theory is to have any application in the real world. Personally I’m interested in real world applications, but I fully acknowledge my ignorance on this particular aspect.
I find a practical way to look at a zero-sum game is to imagine that each of two players must contribute half of the total payoff in order to play. This takes a game that is constant-sum, and makes it zero-sum, and does so in a way that doesn’t break the constant-sumness. In the case of the imbalanced game, because it is not constant-sum it doesn’t reduce to a zero-sum game in this way, remaining a non-zero-sum game with terrible odds for one player, meaning that a rational player won’t opt in if given the option.
If I’m not mistaken, this is generally what is meant when someone refers to a zero-sum game. In chess for instance you enter a competition with your rating (essentially your bet) and the outcome of the game has either a negative or positive (or no) impact on your rating and an opposite impact on your opponent’s rating.(I’m not exactly sure if chess ratings are calculated as exactly zero-sum, but you get the idea). So, the game is zero-sum. Of course there are outside factors that make it beneficial to both players; enjoyment, brain-exercise, socialising etc which may have positive utility on another level, but the game itself and the resulting rating changes are essentially zero-sum.
This is the sense in which I am using the term “zero-sum”, in the most basic sense for someone to win (relative to their starting point, bet or rating) another must lose by an equal amount.
There is probably a more mathematically succinct way of expressing this, but I don’t have those tools at my disposal at present. Again, thanks for your comments. Please don’t feel the need to continue your labours educating me on this topic, I understand that you clearly have a better understanding of game theory than I do, so I appreciate your time. I should probably continue reading further to level up my understanding. Of course if you feel like continuing the floor is yours.
Thanks for your comment, please excuse the delay in getting back, I’m actually busily digesting your response and the various branches of dependencies that comprise it (in terms of links to other concepts). I intend to get back to you with a considered answer, but am enjoying taking my time exploring the ideas you’ve linked to.
See “Zero Sum” is a misnomer, shifting and rescaling of utility functions breaks formulations that simply ask to take a sum of payoffs, but we can rescue the concept to mean that all outcomes/strategies of the game are Pareto efficient.
“Positive sum” seems to be about Kaldor-Hicks improvement, an outcome that in principle admits a subsequent redistribution of resources that would turn the outcome into a Pareto improvement (over some original situation of “not playing”), but there is no commitment or possibly even practical feasibility to actually perform the redistribution. This hypothetical redistribution step takes care of comparing utilities of different players. A whole game/interaction/project would then be “positive-sum” if each outcome/strategy is equivalent via a redistribution to some hypothetical “outcome” that is a Pareto improvement over the status quo of not engaging in the game/interaction/project. In actuality, without the hypothetical redistribution step, some players can end up worse off.
Hi Vladimir, thanks for your input, it has been fascinating going down the rabbit hole of nuance regarding the term “zero-sum”.
I agree that the term is more accurately denoting “constant-sum”, I think this is generally implied by most people using it. There was the interesting “zero-sum” example in the linked article that veered away from “constant-sum” with asymmetrical payoffs, 100,0 or 0,1, meaning that depending on the outcome of the game the total sum would be different. This, to me disqualifies it from being called a zero-sum game, given the common understanding that zero-sum denotes constant-sum. The example seemed to solve the problem by conflating zero-sum and constant-sum and then proceeded to stick to a strict definition of zero-sum, which was confusing. But perhaps I just need to sit with it longer.
To your point about Kaldor-Hicks, yes I guess many positive-sum situations could be described in these terms but I’m really referring to something more general—any situation where the total sum payoff increases regardless of Pareto improvements or promised reimbursement by other means to any party left worse off. For instance if a left-wing government were to increase taxes on the wealth, not offering them any reimbursement, but rather doing this based on the mandate that comes with being democratically elected, then this policy might be positive-sum due to the fact that dollar-for-dollar money makes a bigger difference to a poor person than a rich person, due to diminishing returns on happiness.
I really appreciate your comments, and intend to continue exploring the nuances you’ve raised. I think for a primer on non-zero-sum games, particularly with a site that is focused on practical solutions in the real world rather than pure theory, the more accessible (perhaps less nuanced) definitions I’ve used are probably appropriate.
The point is that the preference order over lotteries characterized by a utility function doesn’t change if you multiply the utility function by a positive value or add a constant to it. Utility function u(x) makes exactly the same choices as utility function 2u(x). If we start with a constant sum-of-utilities game (for two players) and then rescale one of the utilities, the sum will no longer be constant, but the game is still the same. You’d need to take a weighted sum instead to compensate for this change of notation. So the characterization of a game as “constant sum” doesn’t make sense if taken literally, since it doesn’t survive a mere change of notation that doesn’t alter anything about the actual content of the game.
I’m not sure how the game is the same when you add a constant. The game as proposed in the example is clearly different. I can see that multiplication makes no difference, and as such doesn’t make the sum non-constant. I don’t see how asymmetrically changing the parameters is a “mere change in notation”.
By the way, I’m sure you’re entirely correct about this, I just simply don’t see how there is a problem with using the concept of zero-sum understood as constant-sum.
Utility functions are a way of characterizing preference orderings between events. If a preference ordering satisfies certain properties, then there exists a utility function such that its expected value over the events can be used to decide which events are preferred over which other events (see VNM theorem). Utility values are not defined with respect to anything else, they are not money or happiness or resources. In particular, utilities of different players can’t be compared a priori, without bringing in more structure (for example redistribution of resources in the setup of Kaldor-Hicks improvement establishes a way of comparing utilities of players, see the original comment).
If you add a constant to a utility function, its expected value over some event increases by the same constant. So if one event had greater expected utility than another, it would still be the case after you add the constant. This is the sense in which adding constants or multiplying by positive factors makes no difference.
Okay, so I think I get you now, in the imbalanced game, if the payoff is 100 or 1 as in “Zero Sum” is a misnomer, a rational player will still make the same decision, regardless of the imbalance with the other player, given the resulting preference ordering.
However while this imbalance makes no difference to the players’ decisions, it does make a difference to the total payoff, making it non-zero-sum. I’m having difficulty understanding why values such as happiness or resources cannot be substituted for utility—surely at some point this must happen if game theory is to have any application in the real world. Personally I’m interested in real world applications, but I fully acknowledge my ignorance on this particular aspect.
I find a practical way to look at a zero-sum game is to imagine that each of two players must contribute half of the total payoff in order to play. This takes a game that is constant-sum, and makes it zero-sum, and does so in a way that doesn’t break the constant-sumness. In the case of the imbalanced game, because it is not constant-sum it doesn’t reduce to a zero-sum game in this way, remaining a non-zero-sum game with terrible odds for one player, meaning that a rational player won’t opt in if given the option.
If I’m not mistaken, this is generally what is meant when someone refers to a zero-sum game. In chess for instance you enter a competition with your rating (essentially your bet) and the outcome of the game has either a negative or positive (or no) impact on your rating and an opposite impact on your opponent’s rating.(I’m not exactly sure if chess ratings are calculated as exactly zero-sum, but you get the idea). So, the game is zero-sum. Of course there are outside factors that make it beneficial to both players; enjoyment, brain-exercise, socialising etc which may have positive utility on another level, but the game itself and the resulting rating changes are essentially zero-sum.
This is the sense in which I am using the term “zero-sum”, in the most basic sense for someone to win (relative to their starting point, bet or rating) another must lose by an equal amount.
There is probably a more mathematically succinct way of expressing this, but I don’t have those tools at my disposal at present. Again, thanks for your comments. Please don’t feel the need to continue your labours educating me on this topic, I understand that you clearly have a better understanding of game theory than I do, so I appreciate your time. I should probably continue reading further to level up my understanding. Of course if you feel like continuing the floor is yours.
Hi Vladimir,
Thanks for your comment, please excuse the delay in getting back, I’m actually busily digesting your response and the various branches of dependencies that comprise it (in terms of links to other concepts). I intend to get back to you with a considered answer, but am enjoying taking my time exploring the ideas you’ve linked to.